Database Reference
In-Depth Information
In our case, we take for
C
the lfp
enriched (co)complete
category
DB
with the
monoidal product corresponding to the matching database operation
⊗
, with
F
2
=
Σ
D
:
DB
and
F
1
being the hom-functor for a given simple database
A
(i.e., an object in
DB
),
DB
I
(
_
,A)
DB
f
→
◦
K
:
DB
f
→
DB
, where
K
:
DB
f
→
DB
I
is
an inclusion functor. Notice that for any finite database, i.e., object
B
DB
f
,the
DB
I
(K(B),A)
is a hom-object
A
K(B)
of enriched database subcategory
DB
I
.In
this context, we obtain that for any object (also infinite)
A
in
DB
I
(that is, in
DB
),
we have a tensor product
∈
B
∈
DB
f
DB
I
(
_
,A)
K
⊗
P
Σ
D
=
DB
I
K(B),A
⊗
◦
Σ
D
B
B
∈
DB
f
=
DB
I
(B,A)
⊗
Σ
D
B.
This tensorial product comes with a dinatural transformation [
6
],
β
:
S
→
A
,
where
S
DB
O
f
×
DB
and
A
is a constant functor
between the same categories of the functor
S
. Thus, for any given object
A
in
DB
we have a collection of arrows
β
B
:
=
DB
I
(
_
,A)
⊗
Σ
D
_
:
DB
f
→
DB
I
(B,A)
⊗
Σ
D
B
→
A
(for every object
B
∈
DB
f
).
In the standard case of
Set
, which is a (co)complete lfp with the monoidal product
⊗
equal to the Cartesian product
×
, such arrows are
β
B
:
Set
(B,A)
×
Σ
R
(B)
→
A
,
where
B
is a finite set with cardinality
n
, so that
Set
(B,A)
is a set of all tuples
of arity
n
composed of elements of the set
A
, while
Σ
R
(B)
here is
interpreted
as a
set of all
basic n
-ary algebra operations. So that
β
B
is a specification for all basic
algebra operations with arity
n
, and is a function such that for any
n
-ary operation
o
i
∈
=|
B
|
A
n
Σ
R
(B)
and a tuple
(a
1
,...,a
n
)
∈
Set
(B,A)
(where
is an isomorphism
in
Set
) returns the result
β
B
((a
1
,...,a
n
),o
i
)
=
o
i
(a
1
,...,a
n
)
∈
A
. If for a given
B
with
n
,
Σ
R
(B)
is the empty set (that is, we have no
n
-ary operators in
Σ
R
), then
Set
(B,A)
=|
B
|
×
Σ
R
(B)
=
Set
(B,A)
×∅=∅
(
∅
is the initial object in
Set
),
so that
B
∈
DB
f
B
∈
DB
f
&
Σ
R
B
=∅
DB
I
(B,A)
DB
I
(B,A)
⊗
Σ
R
B
⊗
Σ
R
B
.Inthis
case,
β
B
:∅→
A
is an empty function.
In the nonstandard case when, instead of the base category
Set
, another lfp-
enriched (co)complete category is used, as the
DB
category in our case, the
in-
terpretation
for this tensorial product and dinatural transformation
β
is obviously
very different, as we will see in what follows. First of all, the hom-set
DB
I
(B
i
,A)
in the preorder DB category is internalized in a hom-object in
DB
I
as follows:
DB
I
(B
i
,A)
=
in
i
=
TB
if there exists a monomorphism
in
i
:
B
i
→
A
in
DB
I
;
0
otherwise.
Based on the considerations explained before, we obtain that the finitary sig-
nature functor
Σ
D
:
⊥
DB
f
→
DB
has a left Kan extension of
Σ
D
along the cat-
egory inclusion
K
:
DB
f
→
DB
I
[
5
] in the enriched category
DB
,
Lan
K
(Σ
D
)
:
DB
I
→
DB
, and left Kan extension
Lan
J
◦
K
(Σ
D
)
:
DB
→
DB
for the inclusion
functor
J
DB
(this second extension is a direct consequence of the first
one because
J
does not introduce extension for objects, differently from
K
, from
:
DB
I
→
